Asymptotic preserving schemes for nonlinear kinetic equations leading to volume-exclusion chemotaxis in the diffusive limit

TL;DR

This paper develops an asymptotic preserving numerical scheme for nonlinear kinetic equations modeling volume-exclusion chemotaxis, ensuring accuracy, positivity, and energy dissipation in the diffusive limit, validated through biological pattern formation simulations.

Contribution

It introduces a novel micro-macro decomposition-based scheme for nonlinear kinetic equations with density-dependent terms, extending to 2D models and demonstrating key properties.

Findings

The scheme is asymptotic preserving and positivity preserving.

It accurately captures pattern formation in biological systems.

The method is validated through 1D and 2D numerical experiments.

Abstract

In this work we first prove, by formal arguments, that the diffusion limit of nonlinear kinetic equations, where both the transport term and the turning operator are density-dependent, leads to volume-exclusion chemotactic equations. We generalise an asymptotic preserving scheme for such nonlinear kinetic equations based on a micro-macro decomposition. By properly discretizing the nonlinear term implicitly-explicitly in an upwind manner, the scheme produces accurate approximations also in the case of strong chemosensitivity. We show, via detailed calculations, that the scheme presents the following properties: asymptotic preserving, positivity preserving and energy dissipation, which are essential for practical applications. We extend this scheme to two dimensional kinetic models and we validate its efficiency by means of 1D and 2D numerical experiments of pattern formation in biological systems.